On quantum information

We investigate the following generalisation of the entropy of quantum measurement. Let H be an infinite-dimensional separable Hilbert space with a 'density' operator {\rho}, tr {\rho}=1. Let I(P)\in R be defined for any partition P = (P1,...,Pm), P1+ ... +Pm=1H, Pi \in proj H$ and let I(Pi Qj, i \leq m, j \leq n) = I(P) + I(Q) for Q =(Q1,..., Qn), \sum Qj = 1H and PiQj = Qj Pi, tr {\rho} PiQj = tr {\rho} Pi tr {\rho} Q_j (P, Q are physically independent). Assuming some continuity properties we give a general form of generalised information I.